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An object is attached to a spring hanging from the ceiling. The object undergoes simple harmonic motion modeled by the differential equation my'' + ky = 0, where y(t) is the height of the object (relative to its equilibrium position) at time t, m is the mass of the object, and k is the spring constant. (a) Write the phase plane equivalent of the differential equation. The equation should be in dv terms of y,v, and where v = y'. dy dv mu + ky = 0 dy I (b) Integrate the phase plane equivalent equation with respect to y to find an equation relating y to v Write it in the form KE + PE = E k 1/1/2mv ² + 12/27 y ² where KE denotes kinetic energy, PE denotes potential energy, and E denotes total energy. = E (c) Suppose the mass is 5 kg and the spring constant is 3 kg/s2. If the spring is initially stretched 4 meters, held, and released, then determine the total energy and write the resulting equation describing the trajectory of the object in the phase plane. 37 v² + 3y² = 16

An object is attached to a spring hanging from the ceiling. The object undergoes simple harmonic motion modeled by the differential equation my'' + ky = 0, where y(t) is the height of the object (relative to its equilibrium position) at time t, m is the mass of the object, and k is the spring constant. (a) Write the phase plane equivalent of the differential equation. The equation should be in dv terms of y,v, and where v = y'. dy dv mu + ky = 0 dy I (b) Integrate the phase plane equivalent equation with respect to y to find an equation relating y to v Write it in the form KE + PE = E k 1/1/2mv ² + 12/27 y ² where KE denotes kinetic energy, PE denotes potential energy, and E denotes total energy. = E (c) Suppose the mass is 5 kg and the spring constant is 3 kg/s2. If the spring is initially stretched 4 meters, held, and released, then determine the total energy and write the resulting equation describing the trajectory of the object in the phase plane. 37 v² + 3y² = 16

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 59E: According to the solution in Exercise 58 of the differential equation for Newtons law of cooling,...
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An object is attached to a spring hanging from the ceiling. The object undergoes simple harmonic motion modeled by the differential equation my'' + ky = 0, where y(t) is the height of the object (relative to its equilibrium position) at time t, m is the mass of the object, and k is the spring constant. (a) Write the phase plane equivalent of the differential equation. The equation should be in dv = y'. terms of y,v, and where v = dy dv mu + ky = 0 dy I (b) Integrate the phase plane equivalent equation with respect to y to find an equation relating y to v Write it in the form KE + PE = E where KE denotes kinetic energy, PE denotes potential energy, and E denotes total energy. k 1/1/2mv ² + 1/2 y ² = E (c) Suppose the mass is 5 kg and the spring constant is 3 kg/s2. If the spring is initially stretched 4 meters, held, and released, then determine the total energy and write the resulting equation describing the trajectory of the object in the phase plane. ร v²+²²=16
Transcribed Image Text:An object is attached to a spring hanging from the ceiling. The object undergoes simple harmonic motion modeled by the differential equation my'' + ky = 0, where y(t) is the height of the object (relative to its equilibrium position) at time t, m is the mass of the object, and k is the spring constant. (a) Write the phase plane equivalent of the differential equation. The equation should be in dv = y'. terms of y,v, and where v = dy dv mu + ky = 0 dy I (b) Integrate the phase plane equivalent equation with respect to y to find an equation relating y to v Write it in the form KE + PE = E where KE denotes kinetic energy, PE denotes potential energy, and E denotes total energy. k 1/1/2mv ² + 1/2 y ² = E (c) Suppose the mass is 5 kg and the spring constant is 3 kg/s2. If the spring is initially stretched 4 meters, held, and released, then determine the total energy and write the resulting equation describing the trajectory of the object in the phase plane. ร v²+²²=16
(d) Graph the trajectory of the object in the phase plane. -5 -4 -3 -2 -1 Y Clear All Draw: 5 4 3 2 1 = -1 -2 -3 -4 -5 1 2 3 O-XX (e) The graph in part (d) informs us that the solution can be expressed in the form y = C₁ cos(wt) + C₂ sin(wt). Use the assumptions of part (c) to determine the values of C₁ and C₂, and then use the equation of part (c) to determine w. Write the expression for y, describing the position of the object at any time. Keep in mind that the spring is stretched initially, so the initial displacement is negative. 4 X HEA Hint: To determine w, try using a value of t other than 0. 5
Transcribed Image Text:(d) Graph the trajectory of the object in the phase plane. -5 -4 -3 -2 -1 Y Clear All Draw: 5 4 3 2 1 = -1 -2 -3 -4 -5 1 2 3 O-XX (e) The graph in part (d) informs us that the solution can be expressed in the form y = C₁ cos(wt) + C₂ sin(wt). Use the assumptions of part (c) to determine the values of C₁ and C₂, and then use the equation of part (c) to determine w. Write the expression for y, describing the position of the object at any time. Keep in mind that the spring is stretched initially, so the initial displacement is negative. 4 X HEA Hint: To determine w, try using a value of t other than 0. 5
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